本项目拟在半欧氏空间中伪球的Legendrian对偶理论框架下, 研究指标为2的半欧氏空间中的Lorentz光环中的曲线， 超曲面以及高余维子流形等问题。 我们将考虑光锥和光锥之间的对偶以及光锥和指标为2的伪球空间或Anti de Sitter空间的对偶， 从而得到所要研究的子流形的光锥对偶超曲面， 给出光锥对偶超曲面的奇点分类。 进一步，我们将奇点类型进行比较， 以得出子流形的光锥对偶超曲面的奇点集合和子流形的渐屈线（面）的联系。 研究的目标是找到所研究子流形的洛伦兹几何不变量。 对于曲线的研究我们将主要应用相关高度函数族的形变和开折理论。 而对于超曲面和高余维子流形的研究我们将主要应用Legendrian切触理论。

The project will study the singularity theory of the curves, hypersurfaces and submainfolds of high codimension in Lorentz torus in semi-Euclidean space with index two in the framework of the theory of Legendrian dualities between pseudo spheres. We will consider the Legendrian dualities between lightcone and lightcone and the Legendrian dualities between lightcone and Anti de Sitter space or the pseudo sphere with index 2, then we will get the lightcone dual hypersurfaces of the submanifolds. We will study the classifications of singularities of the lightcone dual hypersurfaces. We will compare the singularities of the hypersurfaces to get the connections between the singularities and the evolute of the submanifolds. We will find the Lorentz invariants of the lightcone dual hypersurfaces. For the study of the curves, the main tool is the deformation and unfolding theory for the relative height functions. For the hypersurface and the submanifolds of high codimension, the main tool is Legendrian contact theory.